# Lyapunov Exponents of Two Stochastic Lorenz 63 Systems

## TL;DR

This paper compares the effects of two types of stochastic perturbations, SALT and FD noise, on the Lorenz 63 system, developing a stochastic Lyapunov exponent algorithm and revealing distinct impacts on system volume contraction.

## Contribution

It introduces a stochastic Lyapunov exponent algorithm and analyzes how SALT and FD noise differently affect the Lorenz 63 system's dynamics.

## Key findings

- FD noise alters the sum of Lyapunov exponents significantly.
- SALT noise preserves the original Lorenz 63 Lyapunov sum.
- LA SALT yields a deterministic subsystem isomorphic to Lorenz 63.

## Abstract

Two different types of perturbations of the Lorenz 63 dynamical system for Rayleigh-Benard convection by multiplicative noise -- called stochastic advection by Lie transport (SALT) noise and fluctuation-dissipation (FD) noise -- are found to produce qualitatively different effects, possibly because the total phase-space volume contraction rates are different. In the process of making this comparison between effects of SALT and FD noise on the Lorenz 63 system, a stochastic version of a robust deterministic numerical algorithm for obtaining the individual numerical Lyapunov exponents was developed. With this stochastic version of the algorithm, the value of the sum of the Lyapunov exponents for the FD noise was found to differ significantly from the value of the deterministic Lorenz 63 system, whereas the SALT noise preserves the Lorenz 63 value with high accuracy. The Lagrangian averaged version of the SALT equations (LA SALT) is found to yield a closed deterministic subsystem for the expected solutions which is found to be isomorphic to the original Lorenz 63 dynamical system. The solutions of the closed chaotic subsystem, in turn, drive a linear stochastic system for the fluctuations of the LA SALT solutions around their expected values.